On Friday, July 5, 2002, at 10:54 AM, Bruno Marchal wrote:
> But, perhaps more importantly at this stage I must recall the book
> "Mathematics of Modality" by Robert Goldblatt. It contains fundamental
> papers on which my "quantum" derivation relies. I mentionned it a lot
> some time ago.
> And now that I speak about Goldblatt, because of Tim May who dares
> to refer to algebra, category and topos! I want mention that Goldblatt
> did wrote an excellent introduction to Toposes: "Topoi". (One of the big
> problem in topos theory is which plural chose for the word "topos".
> There
> are two schools: topoi (like Goldblatt), and toposes (like Bar and
> Wells). :)
>
> Goldblatt book on topoi has been heavily attacked by "pure categorically
> minded algebraist like Johnstone for exemple, because there is a remnant
> smell of set theory in topoi. That is true, but that really help for an
> introduction. So, if you want to be introduced to the topos theory,
> Goldblatt Topoi, North Holland editor 19?(I will look at home) is
> perhaps the one.

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Yes, this is an excellent book. It has more of an expositional style
than many books on category and topos theory. It's out of print and
Amazon has been looking for months for a used copy for me. (Amazon can
search for books which become available. I also have them searching for
a copy of Mac Lane and Moerdijk's book on sheaves, logic, and toposes,
also out of print.)
Fortunately, I live near UC Santa Cruz, which has an excellent science
library.
The category and topos theory books I actually _own_ (bought through
Amazon) are:
* Cameron, Peter, "Sets, Logic and Categories, 1998. An undergraduate
level primer on these topics. One chapter on categories. (By the way,
most modern algebra books, e.g., Lang's "Algebra," Fraleigh, Dummitt and
Foote, etc. have introductory chapters on category theory, as this is
the "language" of modern abstract algebra.)
* Lawvere, F. William, Schanuel, Stephen H., "Conceptual Mathematics: A
first introduction to categories," 1997. This is a fantastic
introduction to categorical thinking. The authors are pioneers in topos
theory, but the presentation is suitable for any bright person. There is
not much on applications, and certainly no mention of quantum mechanics
a la Isham, Markopoulou, etc. But the conceptual ideas are profound.
(And this should be read before tackling the "formalistic" presentations
in other books.)
* Pierce, Benjamin, "Basic Category Theory for Computer Scientists,"
1991. A thin (80 pages) book which outlines the basics. Includes
material on compilers, the "Effective" topos of Hyland and others,
cartesian closed categories, etc.
* Mac Lane, Saunders, "Categories for the Working Mathematician, Second
Edition," 1971, 1998. Wow. A dense book by the co-founder of category
theory. As someone said, reading along at 10% comprehension is better
than reading other books at full comprehension. I find the book sort of
dry, on historical and conceptual motivations, but Mac Lane has written
many longer expositions in MAA collections of reminiscences...I just
wish mathematicians would do more of what John Baez in his papers: show
the reader the motivations.
(Much of mathematical writing came out of the tradition of "lecture
notes." In fact, the leading publisher of mathematics, Springer-Verlag,
calls their series "Lecture Notes," or, more recently, "Graduate Texts."
Brilliant mathematicians like Emil Artin and Emmy Noether would have
their lectures on algebra transcribed by grad students or post-docs,
like Van der Waerden, who would then republish the notes as "Moderne
Algebra," the first of the "groups-rings-fields" modern algebra books.
Which is why one of E. Artin's students, Lang, writes so many dry books!
These books are often very short on pictures or diagrams, very short on
segues and motivations. It's as if all of what a good teacher would do
in class, with drawings on blackboards, with historical asides, with
mentions of how material ties in with material already covered, with
mention of open research problems and unexplored territory...it's as if
all this material is just left out of these texts. Too bad.)
* Lambek, J., Scott, P.J., "Introduction to higher order categorical
logic," 1986. Way too advanced for me at this point. So no comments on
content. But it's useful to glance at topics so as to get some idea of
where things are going (part of the issue of motivation I raised above).
* Taylor, Paul, "Practical Foundations of Mathematics," 1999. Another
advanced book, covering logic, recursive function theory, cartesian
closed categories, and a lot of the second half I can't comment on. A
wonderful browsing book, as he has lots of tidbits and asides.
There are 3-4 other books I'd like to get, including the Goldblatt book
(he is giving permission to Xerox his book, so I may do that), the Mac
Lane and Moerdijk book, and a few others. Peter Johnstone wrote the
defining book on toposes in 1977...long-since out of print and
long-since overtaken by newer results. Ah, but he is about to have his
massive 3-volume set of books on topos theory published:
Here's John Baez's summary in his Week 180 column:
"2) Peter Johnstone, Sketches of an Elephant: a Topos Theory Compendium,
Cambridge U. Press. Volume 1, comprising Part A: Toposes as Categories,
and Part B: 2-categorical Aspects of Topos Theory, 720 pages, to appear
in June 2002. Volume 2, comprising Part C: Toposes as Spaces, and Part
D: Toposes as Theories, 880 pages, to appear in June 2002. Volume 3,
comprising Part E: Homotopy and Cohomology, and Part F: Toposes as
Mathematical Universes, in preparation.
"I can't wait to dig into this. A topos is a kind of generalization of
the universe of set theory that we all know and love, but topos theory
is really a wonderful way to unify and generalize vast swathes of
mathematics - you could say it's the way that logic and topology merge
when you take category theory seriously. I've really just begun to get a
glimmering of what it's all about, so I'm curious to see Johnstone's
overall view of the subject. "
(end of John Baez's comments)
The first two volumes are due this month or next, according to Oxford
University Press (_not_ Cambridge!) and Amazon. Cost for the two is a
whopping $295. But the books are 750 and 850 pages, respectively.
I'm am steeling myself to buying them. A lot of money, but this stuff is
more entertaining to me than spending the same amount for 1-2 nights in
a hotel, or lots of other things people spend their money on. And
obviously 1500 pages is a lot of reading!
And to better understand these things, I've been brushing up on my math
background. A lot of algebra texts (mentioned above), topology (Munkres,
Hocking and Young, Alexandroff, various Dover editions), and algebraic
topology (Massey, Bredon, Fulton, etc.). My background is mostly
physics, but I fortunately had some good exposure to analysis and
measure theory, the stuff that can provide the assumed "mathematical
maturity" for further study. I wish I'd spent more time studying this
stuff...but wishing about changes in the past is pointless.
I'm here now, in my one and only present, and this category and topos
theory is turning out to be enjoyable and stimulating as a goal unto
itself, and as a tool for, I think, better understanding things I want
to understand.
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks